91影视

For a given vector field $F(x,y,z)$and simple closed curve $C$, traversed counterclockwise to a chosen normal vector $n$, the circulation of $F(x,y,z)$around $C$measures the rotation of the fluid about $C$in the direction counterclockwise to the aforementioned chosen normal vector and is defined to be $CF(x,y,z)脳dr.$ Find the circulation of the given vector field around $C$in Exercises $37and38$.

$F(x,y,z)=i+j+k,$and $C$is the curve of intersection of the plane $胃=\frac{蟺}{4}$or $胃=\frac{5蟺}{4}$and the unit sphere.

Directly compute (i.e., without using Green鈥檚 Theorem) ${鈭�}_C鈥�\mathbf{F}(x,y).dr$, where $\mathbf{F}(x,y)=\left(2^x+y\right)\mathbf{i}+(2x鈭�y)\mathbf{j}$

and C is the triangle described by x = 0, y = 0, and y = 1 鈭� x, traversed counterclockwise.

Show that the vector fields in Exercises 33鈥�40 are not conservative.

$\mathbf{F}(x,y,z)=2\mathbf{i}-z\mathbf{j}+{\mathrm{e}}^{\mathrm{yz}}\mathbf{k}$

$\mathbf{F}(x,y,z)=鈭�xz\mathbf{i}鈭�yz\mathbf{j}+z^2\mathbf{k}$, where S is the cone with equation $z=\sqrt{x^2+y^2}$between $z=2,4$, with n pointing outwards.

Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37鈥�44. Otherwise, show that the vector field is not conservative.

$\mathbf{F}(x,y)=(2x\ln 2,2y)$, with C the straight line segment from $(3,7)\mathrm{to}(0,1)$.

For a given vector field $F(x,y,z)$and simple closed curve $C$, traversed counterclockwise to a chosen normal vector $n$, the circulation of $F(x,y,z)$around $C$measures the rotation of the fluid about $C$in the direction counterclockwise to the aforementioned chosen normal vector and is defined to be $CF(x,y,z)脳dr$. Find the circulation of the given vector field around $C$in Exercises $37and38$.

, and $C$is the intersection of the cone $z=\sqrt{x^2+y^2}$and the unit sphere.

Show that the vector fields in Exercises 33鈥�40 are not conservative.

$\mathbf{F}(x,y,z)=\tan \left(yz\right)\mathbf{i}+\left(xzse{c}^2\right(yz)鈭�2)\mathbf{j}+4z^3\mathbf{k}$

$a$Use Green鈥檚 Theorem to evaluate the line integral in Exercise 37

$\mathbf{F}(x,y,z)=鉄�yz,xz,xy鉄�$, where S is the portion of the saddle determined by$z=x^2鈭�y^2$ that lies above the region in thexy-plane bounded by the x-axis and the parabola with equation$y=1鈭�x^2$.

Consider once again the notion of the rotation of a vector field. If a vector field $F(x,y,z)$has curl $F=0$at a point $P$, then the field is said to be irrotational at that point. Show that the fields in Exercises $39鈥�42$are irrotational at the given points.

$\mathbf{F}(x,y,z)=\left\{鈭�\sin 鈦�x,3y^3,4z+12\right\},P=(2,3,4)$